Differential Equations

The Geometric View of ODEs

01Direction Fields and the Geometric View of ODEs 02Integral Curves and the Isocline Method 03Perpendicular Isoclines: The Circles of y' = -x/y 04Isocline Corridors and a Trap for Solutions 05Existence and Uniqueness: One Curve Through a Point 06Where Existence and Uniqueness Fail Problem set0/10 Problem set 20/10 MIT problem set0/3 Practice∞

Numerical Methods

01Euler's Method and Its Three Equations 02Euler's Method by Hand: Building the Table 03Convexity and the Sign of Euler's Error 04First Order Error: Halve the Step, Halve the Error 05Improved Euler (RK2): Averaging Two Slopes 06Runge-Kutta 4 and the Price of Accuracy 07Singular Points: Where Every Method Fails Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

First-Order Linear Equations

01First Order Linear: Standard Form and the Sign of p 02The Conduction-Diffusion Model: Newton's Law of Cooling 03The Integrating Factor and the Four-Step Method 04Integrating Factors: Two Mistakes and a Trick Question 05Constant Coefficients: Steady State and Transient Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

The Integrating Factor and the Four-Step Method

What forces the integrating factor to be $u = e^{\int p\,dx}$, and why does no arbitrary constant survive the derivation?

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Your summary note

1. The integrating factor $u=e^{\int p\,dx}$
  Multiply $y'+py=q$ by $u$ , match $(uy)'=uy'+u'y$ from the product rule to force $u'=pu$ , separate variables, and state $u=e^{\int p\,dx}$ with no arbitrary constant.
2. The four steps of the method
  Write the four steps in order (standard linear form, the integrating factor, multiplying both sides, integration) and record the wrong-sign trap from reading $p$ off $y'=-py+q$ .
3. The worked example $xy'-y=x^3$
  Carry it through: divide by $x$ for standard form, simplify $e^{-\ln x}$ to $1/x$ , write $\left(\frac{1}{x}y\right)'=x$ and verify it with the product rule, then integrate to $y=\frac{x^3}{2}+cx$ .
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